**3.3 Mathematical modeling**

Also, the Herchel-Bulkley model indicated that reactor fluid A performed as a pseudo-Newtonian fluid called Bingham plastic, since the yield stress-value was > 0 (0.24 Pa) and a flow behaviour index of 1.06 (Table 4). Results obtained by the Ostwald and Bingham models confirmed a Bingham plastic behaviour of reactor A. However, since the 0-value was almost 0 and the n-value 1 it was also closely performing as a Newtonian fluid which is consistent with the flow curve appearance (Fig. 2). However, when studying the viscosity curve (Fig. 4) the results showed an initial viscosity decrease and then a constant viscosity indicating a pseudo-Newtonian fluid behaviour.

The Herschel-Bulkley and Ostwald models both indicated a pseudoplastic behaviour of reactor D, since the 0-value was 0 and n < 1 (Table 4). The Bingham model gave a yield stress of 0.33 Pa which did not indicate Newtonian or Bingham plastic behaviour. Thus, the common results for reactor D strongest indicate a pseudoplastic fluid behaviour.

Reactor B was hard to define also when modelling the rheogram data values of figure 4. The regression values were low for all three mathematical models (Table 4). However, the Herschel-Bulkely model had a flow behaviour index n>1 indicating that the fluid acted as a shear thickening (dilatant) fluid, but the Ostwald and Bingham models indicated pseudoplastic and Bingham plastic behaviours, respectively. When the static yield stress appeared in the reactor B rheogram (Figures 2 and 4), the flow behaviour index showed shear thickening fluid behaviour (n=3.4) and a limit viscosity of 8 mPa\*s. This also

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corresponded to a low consistency value (5\*10-10). At the static yield stress of 24 Pa (Fig. 5), the flow behaviour index showed shear thickening fluid behaviour (n=1.41) and a limit viscosity of 22 mPa\*s. This also corresponded to a low consistency value (5\*10-4). As soon as the fluid was measured again, n decreased (0.70) showing a pseduoplastic behaviour and K increased (0.11) indicating that the consistency of the reactor material was higher. The limit viscosity was 17 mPa\*s. These results showing a time dependency and structure recovery strengthen the arguments for a thixotropic fluid behaviour of reactor B. Once the stirring has ended and the fluid was at rest, the fluid structure starts to rebuild. Therefore, the viscosity become time dependent. This information is important to consider for biogas reactor performance, e.g. when applying semi-continuous mixing.


Table 4. The results obtained from mathematical modelling of rheogram data of fluids from reactors A−E. 0: yield stress (Pa); n: flow behaviour index; : Consistency index; R2: regression coefficient.

Also, fluids from reactor C and E were hard to define from modelling of the rheogram data because they gave indications for fluids being between pseudoplastic and Bingham plastic behaviours, i.e. the 0-values were >0 (2.89 and 2.38) and n <1 (Table 4).
